We introduce and study a new class of generalized inverses in rings. An element a in a ring R has p-Hirano inverse if there exists b ∈ R such that bab = b, b ∈ comm 2 (a), (a 2 − ab) k ∈ J(R) for some k ∈ N. We prove that a ∈ R has p-Hirano inverse if and only if there exists p = p 2 ∈ comm 2 (a) such that (a 2 − p) k ∈ J(R) for some k ∈ N. Multiplicative and additive properties for such generalized inverses are thereby obtained. We then completely determine when a 2 × 2 matrix over local rings has p-Hirano inverse